#411266

order	512
nilpotency class	3
isoclinism	isoclinic class of #399126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(2,3)(5,6,10,15,11,16,18,34)(7,17,29,32,21,12,22,41)(8,13,25,36,26,37,44,58)(9,14,38,59,27,39,28,40)(19,42,51,33,46,30,23,49)(20,31,47,56,48,43,54,63)(24,35,60,68,52,61,53,62)(45,55,50,57,65,64,67,66);(1,2,4,3)(5,7,18,22,11,21,10,29)(6,12,15,41,16,17,34,32)(8,23,44,46,26,51,25,19)(9,20,38,47,27,48,28,54)(13,33,36,30,37,49,58,42)(14,31,40,63,39,43,59,56)(24,50,60,65,52,67,53,45)(35,57,62,55,61,66,68,64);(1,2,4,3)(5,8,18,44,11,26,10,25)(6,13,34,58,16,37,15,36)(7,19,29,51,21,46,22,23)(9,24,38,60,27,52,28,53)(12,30,41,49,17,42,32,33)(14,35,59,68,39,61,40,62)(20,45,54,67,48,65,47,50)(31,55,63,66,43,64,56,57);(1,3,4,2)(5,9)(6,14)(7,20)(8,24)(10,28)(11,27)(12,31)(13,35)(15,40)(16,39)(17,43)(18,38)(19,45)(21,48)(22,47)(23,50)(25,53)(26,52)(29,54)(30,55)(32,56)(33,57)(34,59)(36,62)(37,61)(41,63)(42,64)(44,60)(46,65)(49,66)(51,67)(58,68)

Polycyclic presentation:

2^9|a^2=efg,b^2=gh,c^2=fgi,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318462);